Protected Convalescence: Night Nurse can use Support. After that action resolves, modify her target’s defense value by +1 until the beginning of your next turn.

Does the target character need to heal to get the bonus to its defense value?
No. In fact, the roll to hit with Support doesn't even need to be successful. After the action resolves (successful or not), the target of the action will have its defense modified.

If I use the feat Triage to heal multiple characters will they all receive the +1 bonus?
No. According to the description, Triage works kind of like Energy Explosion: you choose a character to target and, if they are successfully hit, then you can also choose to heal characters adjacent to that target if the roll was high enough to hit them, too. However, since Protected Convalescence modifies the target's defense value, and Triage only ever refers to the intial character as the target, any other characters healed are not targets and will not receive the modification to defense value.